Thermodynamics of Gradient Driven Transport: Application to Single-Particle Tracking R. Dean Astumian* and Ross Brody Department of Physics, UniVersity of Maine, Orono, Maine 04469 ReceiVed: April 23, 2009; ReVised Manuscript ReceiVed: June 25, 2009 Single-particle tracking techniques allow measurements of the trajectories of individual particles (and sometimes single molecules) as they move. Here, we show, in accord with an equality derived by Bier et al. (Phys. ReV. E 1999, 59, 6422-6432), that even in the presence of gradients of pressure, temperature, electric field strength, and so forth, the ratio of the probabilities for forward and backward trajectories between any two points is given by the exponential of the difference between the particle’s free energy at the start and end points of the trajectory, that is, P(x a f x b , Δt) ) P(x b f x a , Δt) exp(-ΔG/k B T), where ΔG ) G(x b ) - G(x a ). Thus, experimental approaches based on single-particle tracking can be used to map the free-energy landscape for transport of a particle without reference to whether the overall system is, or is not, in thermodynamic equilibrium. 1. Introduction The Gibb’s free energy of a particle or macromolecule in solution is a well-defined equilibrium property that depends on the extensive parameters of the particle, its molecular volume, dipole moment, entropy, and so forth, and on the intensive parameters of the solution, the temperature, pressure, electric field intensity, and so forth. The differential of the free energy dG, the change due to a very small change in the intensive parameters, can be expressed as the sum of products 1 where the θ i and F i are the canonically conjugate extensive and intensive thermodynamic parameters of the system, respectively. Examples include the familiar pairs of thermodynamic variables SdT, where S is the particle entropy and T is the temperature, and Vdp, where V is the molecular volume and p is the pressure, as well as pairs of mechanical parameters md(gh), where m is the effective mass, g is the acceleration due to gravity, and h is the height, and ldf, where l is length in the direction of an external force f, and pairs of electromagnetic parameters μdE, where μ is the molecular polarization and E is the electric field, and qdφ where q is the charge and φ is the electrical potential. The free-energy change of a particle ΔG 1,2 ) G(F(2)) - G(F(1)) and ΔG 2,3 ) G(F(3)) - G(F(2)) due to reversible but nonin- finitsemal changes of the intensive parameters of the system (Figure 1a) F(1) f F(2) f F(3) (Figure 1a) is, in general, a nonlinear function of the intensive parameters, that is, G(F(2)) - G(F(1)) * G(F(3)) - G(F(2)) even if F(2) - F(1) ) F(3) - F(2). In the presence of a spatial gradient ∇F of one of the intensive parameters (e.g., temperature, electric field strength, pressure, and so forth), a particle will move, tending to seek the minimum of its free energy. In Figure 1b, a diffusive trajectory for a particle is shown, where the particle is at some position x a at time t i , and by Brownian motion, it arrives at the position x b at some time t f , where the vector x ) (x, y) specifies the position in a two-dimensional Cartesian coordinate system. Onsager 2 postulated that the average current in an ensemble of particles is related to the gradient of the “thermodynamic potential” by a linear relation, RJ ) ∇F. Onsager and Machlup 3 extended this picture to include fluctuations due to thermal noise in terms of the overdamped Langevin equation RR ˙ ) ∇F + ε(t), where R ˙ is the rate of change of some thermodynamic variable. 2. Results and Discussion 2.1. Onsager-Machlup Theory for Single-Particle Tra- jectories. Here, following Onsager and Machlup, we propose that the local velocity x ˙ for a single particle can be related to the equilibrium free energy by a similar Langevin equation, reflecting a locally linear relation between the velocity and the thermodynamic force that caused it * To whom correspondence should be addressed. E-mail: astumian@ maine.edu. dG ) ∑ i θ i dF i (1) Figure 1. (a) Schematic illustration of a reversible (quasi-static) process in which an intensive parameter is changed, F(1) f F(2) f F(3), leading to the change in the free-energy of a particle G[F(1)] f G[F(2)] f G[F(3)]. For large changes of the intensive parameter, G[F] is a nonlinear function of F, but the overall process is well described by equilibrium thermodynamics. (b) Illustration of a single-particle trajec- tory in a gradient ∇F. We prove theoretically and show experimentally that the ratio of the probabilities for forward and backward trajectories is the exponential of the free-energy difference at the end points, P[x a f x b , Δt] ) P[x b f x a , Δt] exp(-ΔG/k B T). J. Phys. Chem. B 2009, 113, 11459–11462 11459 10.1021/jp903746j CCC: $40.75  2009 American Chemical Society Published on Web 07/28/2009